EMÜ 221 Introduction to Optimization and Modeling Ceren Tuncer Şakar 2015-2016 Fall Week 2
2 Feasible Region The feasible region of an LP is the set of all points satisfying all the LP’s constraints and sign restrictions. Feasible Solution Any solution that satisfies all constraints Infeasible Solution A solution that is not feasible is said to be infeasible Optimal Solution The feasible soluton that gives the best objective function value
Slack and Surplus Variables A linear program in which all the variables are non- negative and all the constraints are equalities is said to be in standard form . Standard form is attained by adding slack variables to "less than or equal to" type constraints, and by subtracting surplus variables from "greater than or equal to" type constraints. Slack and surplus variables represent the difference between the left and right sides of the constraints. Slack and surplus variables have objective function coefficients equal to 0. 3
4 Max z = 3x 1 + 2x 2 + 0 s 1 + 0s 2 + 0s 3 s.t. 2 x 1 + x 2 + s 1 = 100 x 1 + x 2 + s 2 = 80 x 1 +s 3 = 40 x 1 , x 2 , s 1 , s 2 , s 3 ≥ 0 Standard Form
5 Graphical Solution to a 2-Variable LP
6 Graphical Solution to a 2-Variable LP X1 X2 10 20 40 50 60 80 finishing constraint carpentry constraint demand constraint z = 60 z = 100 z = 180 Feasible Region G A B C D E F H From figure, we see that the set of points satisfying the Giapetto LP is bounded by the five sided polygon DGFEH. Any point on or in the interior of this polygon (the shade area) is in the feasible region.
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- Fall '15
- Optimization, Constraint, feasible region, Candidate solution, Giapetto LP